## Rational Curves on Algebraic VarietiesThe aim of this book is to provide an introduction to the structure theory of higher dimensional algebraic varieties by studying the geometry of curves, especially rational curves, on varieties. The main applications are in the study of Fano varieties and of related varieties with lots of rational curves on them. This Ergebnisse volume provides the first systematic introduction to this field of study. The book contains a large number of examples and exercises which serve to illustrate the range of the methods and also lead to many open questions of current research. |

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algebraic algebraically closed field ample apply Assume assumption birational called Cartier divisor char characteristic zero Chow Claim complete complete intersection compute connected consider construction contained Corollary corresponding cover defined definition deformation denote dimension dimensional dominant embedding equivalent examples Exercise extension Fano varieties fiber finite fixed flat geometrically given gives hence Hilbert holds hypersurface ideal implies induction intersection irreducible component isomorphism least Lemma Let f Let X line bundle locally morphism natural normal Notation obtain particular projective projective variety Proof proper properties Proposition prove quotient rational curves rationally connected reduced relation resp result ruled satisfies scheme semi separably sequence sheaf Show singular smooth space Spec subscheme subset subvariety sufficient surface Theorem theory uniruled

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Page 3 - S, etc. to determine the thermodynamic state. It is assumed that the reader of this book is familiar with these terms. Here we just briefly mention some of the important relations for these quantities which will be used in this book. The...

Page 316 - VG Sarkisov, Birational automorphisms of conic bundles, Math. USSR Izv. 17 (1981), 177-202. [Sarkisov82] VG Sarkisov, On the structure of conic bundles, Math. USSR Izv. 20 (1982), 355-390. [Segre43] B. Segre, A note on arithmetical properties of cubic surfaces, J. London Math.